Fourier transforms of measures invariant under the Gauss map

Normally measures with decaying Fourier transform arise in random models. For example, Brownian motion gives several natural examples. We will now study how dynamical properties of a measure affect its Fourier transform. In general, answer to such a question can be quite simple: for example, the only invariant measure for the doubling map with Fourier decay at infinity is the Lebesgue measure. However, for the Gauss map describing the evolution of continued fraction coefficients the question is more involved and we prove that a wide class of statistically rich invariant measures (Gibbs measures) have power decay for the Fourier transform. The conditions we present are satisfied for several fractal measures such as the conformal measure (Hausdorff measure) on the set of $n$-badly approximable numbers. Moreover, by invoking the Davenport-Erdos-LeVeque criterion on equidistribution, we give some new equidistribution results for Gibbs measures. This is a joint work with Thomas Jordan (Bristol).